The age of painting was determined from the decay kinetics of the radioactive Carbon -14 present in the painting sample.
Given that the half life of Carbon-14 is 5730 years.
Radioactive decay reactions follow first order rate kinetics.
Calculating the decay constant from half life:
λ[tex]= \frac{0.693}{t_{1/2} }[/tex]
= [tex]\frac{0.693}{5730 yr}[/tex] = [tex]1.21*10^{-4}yr^{-1}[/tex]
Setting up the radioactive rate equation:
[tex]ln\frac{A_{t} }{A_{0} } =-kt[/tex]
Where [tex]A_{t} = Activity after time t = 0.80microCi[/tex]
[tex]A_{t} = initial activity = 6.4microCi[/tex]
k = decay constant = [tex]1.21*10^{-4}yr^{-1}[/tex]
[tex]ln\frac{0.80uCi}{6.4uCi} =-(1.21*10^{-4}yr^{-1})t[/tex]
ln 0.125 = [tex]-(1.21*10^{-4}yr^{-1})t[/tex]
-2.079=[tex]-(1.21*10^{-4}yr^{-1})t[/tex]
t=[tex]\frac{2.07944}{1.21*10^{-4} } yr[/tex]
= 17185 years
t = 17185 years
Therefore age of the painting based in the radiocarbon -14 dating studies is 17185 years
